The Green’s function is a concept developed by George Green in the XIXth century. It is useful to solve inhomogenous differential equations such as :

where the functions f and g are defined on an interval.

**GREEN’S FUNCTION**

More generally, consider an equation with a linear differential operator L :

**Green’s function**

A Green’s function of a linear operator L is any solution of the equation

where is the Dirac delta-function.

The original equation can be rewritten as follows :

In this way, the Green’s function plays the role of an integral kernel in the solution of the equation .

**INHOMOGENEOUS LINEAR EQUATIONS**

A first order inhomogenous linear equation is an equation

It can be solved with initial condition :

**GREEN’S FUNCTION OF A INHOMOGENOUS LINEAR EQUATION**

*What is the Green’s function of a first order inhomogenous linear equation ?*

The Green’s function satisfies the equation

Then, we apply the solution defined above of a inhomogenous linear equation with the condition

**EXAMPLE**

We illustrate the Green’s function on the following equation

The Green’s function is computed as

It is graphically represented at the beginning of the post.

We deduce the solution

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